It sounds as though your problem was a somewhat loose tyre not properly seated. It may have been loose enough that it couldn't be properly seated, but in that case I'd expect it to pop off and the tube explode while you were pumping it up for the first time. You may also be running the tyres at too low a pressure, with a skinny tyre like that I'd expect them to need 70psi or more.
So, seating a new tyre:
- fit the uninflated tyre and tube as you normally do
- pump it up to 10-15psi - just enough that there's air in there but it's still very spongy
- ideally spin the wheel, but just looking around it also works. Look for bulges or dips in the tyre sidewall relative to the rim, ideally using a line around the tyre (a moulding line or the reflective strip).
- when you see a dip down into the rim, grab the tyre firmly on both sides and rock your hand to pull that bulge up into the correct position.
- when you see a bulge upward you need to pull the rest of the tyre down. Start 1 few inches past the bulge and rock the tyre up (as above), then repeat this around the tyre until you're back at the bulge (which should be gone by now)
- repeat on the other side of the tyre.
Steps 4 and 5 are easy to skip, and the reason why you want very little air in that tyre. The "explode on first inflation" thing happens occasionally in bike shops, and it's a chance for everyone else to laugh at whoever did it. After getting over the shock of the loud noise!
Normally this is just a quick eyeball check of both sides of the tyre, and I'll often skip it with tyres that are tight on the rim. If it's hard to get on, it'll usually seat firmly as soon as air goes in. But loose tyres I'm very careful of (see "loud noise" above).
(edit to add)
My impression is that many (most?) cyclists I know go to a local bike shop and talk to people, see what's available, and buy something they can actually touch. At least once, anyway, before going "I can get these much cheaper online". Having decided on a brand or brands, and a preference for fat/slick/kevlar-studded and so on, they then grind through a few options over the next few thousand kilometres of riding before developing either very strong preferences (like the bike geeks here), or a complete lack of interest. Working in a bike shop I really did get people who brought bikes in and said "tyre. flat. replace. bye" then walked out, leaving me holding a bike going "can I at least get your phone number?"
Availability is also important. If you're going to try some tyres with a view to picking one you like, pick a manufacturer that's been around for a while and doesn't change their darn tyres every year. Otherwise every time you buy a tyre you have to work out which one best matches the one you bought last year. If you're buying online, try to pick a store that keeps stuff in stock. It's annoying buying something only to wait a month while it gets shipped from a warehouse in China when you chose a "local" online store.
So, my preference is for puncture resistant to very, very puncture resistant slicks, with more puncture resistance on my commuter and less on my touring bike (where rolling resistance is more important). On my velomobile I went for extreme puncture resistance because going much faster also meant that the time taken to stop and fix a puncture stood out a lot more. Spending 10 minutes fixing a puncture on a 20km ride that should have taken under 30 minutes is more annoying than I expected, so after it happened once I went for serious puncture resistance. I also prefer a brand (Schwalbe) and apart from a few older Maxxis tyres that haven't worn out yet, all... 6... bikes in our shed have Scwalbe tyres. Marathon Plus through to Marathons, with Duranos on my 700c "lazy bike" (takes less effort to go faster than by other bikes, therefore "lazy")
Somewhere there's a photo of a 3" nail sticking through a marathon plus on my bike. Not, I emphasise, puncturing the marathon plus, just "zombie tyre" poking through it, with one side of the nail head ground flat where it had hit the ground every time the wheel went round. That's freakishly unusual, but it's one reason why I like the marathon plus tyres.